Beyond the spherical cow

One of the most well-known approximations in physics is the spherical cow. For generations, physicists have treated cows as spheres, a simplifying assumption to make the math easier. However, this approximation renders several problems unsolvable.

Benjamin Lehmann improved upon the utility of this classic approximation by treating the spherical cow as simply the first term of a multipole expansion. This allowed him to resolve higher order effects that were previously impossible to explore.

“If you just take the leading order term for the cow’s shape, you lose out on all the effects that completely vanish at the monopole order,” said Lehmann. “The first thing that my mind went to is gravitational waves, because gravitational waves famously only come from the quadrupole and higher moments.”

Using his multipole expansion, Lehmann calculated the rate at which a spinning cow loses energy to gravitational waves — although a complete treatment would also include corrections due to the loss of angular moo-mentum.

He also addressed the problem of cow tipping, calculating the minimum force required to overturn the cow. A spherical approximation disagrees badly with experiments, which clearly indicate the force is not vanishingly small. By including higher order approximations, Lehmann was able to calculate the optimal force and point of application to overcome the torque due to the cow’s own weight.

Lehmann emphasized that these are only a handful of the many cow-related problems that can be solved through a multipole expansion.

“I think one of the interesting questions is bovine deformability. If I subject a cow to an external potential that is going to deform the cow a little bit, to what extent am I going to induce contributions to these higher moments?” said Lehmann. “In astrophysical contexts, these are called Love numbers. And you could imagine studying the Love numbers of a cow to understand exactly how squishy it is.”

Source: “Higher multipoles of the cow,” by Benjamin V. Lehmann, American Journal of Physics (2026). The article can be accessed at https://doi.org/10.1119/5.0287411 .